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Trigonometry - Calculating Sides and Angles in Right-Angled Triangles

Grade 8IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Identification of sides in a right-angled triangle: Hypotenuse (longest side across from the right angle), Opposite (side across from the target angle), and Adjacent (side next to the target angle).

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SOH CAH TOA mnemonic for remembering the three primary trigonometric ratios.

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Using trigonometric ratios to find a missing side length when one angle and one side are known.

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Using inverse trigonometric functions (sin⁻¹, cos⁻¹, tan⁻¹) to find a missing angle when two sides are known.

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Ensuring the scientific calculator is set to 'Degrees' (DEG) mode for calculations.

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The relationship between Pythagoras' Theorem and Trigonometry in solving right-angled triangles.

📐Formulae

sin⁡(θ)=OppositeHypotenuse\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}

cos⁡(θ)=AdjacentHypotenuse\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}

tan⁡(θ)=OppositeAdjacent\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}

a2+b2=c2a^2 + b^2 = c^2 (Pythagoras' Theorem)

θ=sin⁡−1(OppositeHypotenuse)\theta = \sin^{-1}\left(\frac{\text{Opposite}}{\text{Hypotenuse}}\right)

💡Examples

Problem 1:

In a right-angled triangle, the hypotenuse is 12 cm and one angle is 35°. Calculate the length of the side opposite to the 35° angle. Give your answer to 2 decimal places.

Solution:

6.88 cm

Explanation:

  1. Identify the given information: Angle θ=35∘\theta = 35^\circ, Hypotenuse = 12 cm. We need the Opposite side. 2. Choose the ratio: SOH uses Opposite and Hypotenuse. 3. Set up the equation: sin⁡(35∘)=x12\sin(35^\circ) = \frac{x}{12}. 4. Solve for xx: x=12×sin⁡(35∘)≈12×0.573576=6.8829...x = 12 \times \sin(35^\circ) \approx 12 \times 0.573576 = 6.8829... 5. Round to 2 decimal places.

Problem 2:

A right-angled triangle has an adjacent side of 7 cm and an opposite side of 5 cm relative to an angle θ\theta. Find the value of θ\theta to 1 decimal place.

Solution:

35.5°

Explanation:

  1. Identify the given information: Opposite = 5 cm, Adjacent = 7 cm. 2. Choose the ratio: TOA uses Opposite and Adjacent. 3. Set up the equation: tan⁡(θ)=57\tan(\theta) = \frac{5}{7}. 4. Use the inverse tangent function: θ=tan⁡−1(57)\theta = \tan^{-1}(\frac{5}{7}). 5. Calculate: θ≈tan⁡−1(0.7142)=35.537...\theta \approx \tan^{-1}(0.7142) = 35.537... 6. Round to 1 decimal place.